Abstract

We discuss the construction of proper widening operators on several
weakly-relational numeric abstractions. Our proposal differs from
previous ones in that we actually consider the semantic abstract domains,
whose elements are geometric shapes, instead of the (more concrete)
syntactic abstract domains of constraint networks and matrices.
Since the closure by entailment operator preserves geometric shapes,
but not their syntactic expressions, our widenings are immune from
the divergence issues that could be faced by the previous approaches
when interleaving the applications of widening and closure.
The new widenings, which are variations of the standard widening
for convex polyhedra defined by Cousot and Halbwachs, can be made
as precise as the previous proposals working on the syntactic domains.
The implementation of each new widening relies on the availability of
an effective reduction procedure for the considered constraint description:
we provide such an algorithm for the domain of octagonal shapes.